Hua–Huang–Li conjecture on independent random waves for Hecke–Maass cusp forms

Let $J\geq2$ be fixed, and let $f_j$, $1\leq j\leq J$, be pairwise orthogonal $L^2$-normalized Hecke–Maass cusp forms on $\mathbb{X}=\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, with spectral parameters $t_{f_j}$. For integers $a_j$, define

$$c_n=\begin{cases} \left(\frac{3}{\pi}\right)^{n/2}(n-1)!!&\text{if $n$ is even},\\ 0&\text{if $n$ is odd}. \end{cases}$$

Hua–Huang–Li conjecture. The powers $\{f_j^{a_j}\}_{j=1}^J$ are statistically independent: for every $\psi\in\mathcal{C}_c^\infty(\mathbb{X})$,

$$\int_{\mathbb{X}}\psi(z)\prod_{j=1}^J f_j(z)^{a_j}\,\mathrm{d}\mu z =\prod_{j=1}^Jc_{a_j}\int_{\mathbb{X}}\psi(z)\,\mathrm{d}\mu z+o(1)$$

as $\min(t_{f_1},\ldots,t_{f_J})\to\infty$. This is a joint value-distribution conjecture asserting that distinct Hecke–Maass cusp forms behave as independent random waves. The source presents it as a conjecture; no resolution is supplied here.